A body has maximum range `R_1` when projected up the plane. The same body when projected down the inclined plane, it has maximum range `R_2`. Find the maximum horizontal range. Assume equal speed of projection in each case and the body is projected onto the inclined plane in the line of the greatest slope.

Explanation

To solve the problem of finding the maximum horizontal range when a body is projected up and down an inclined plane, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a body projected on an inclined plane at an angle θ with the horizontal. We denote the maximum range when projected up the plane as \( R_1 \) and when projected down the plane as \( R_2 \). We need to find the maximum horizontal range \( R \).

2. **Formulas for Maximum Range**:
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